The IMSL_QUADPROG function solves a quadratic programming (QP) problem subject to linear equality or inequality constraints.

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The IMSL_QUADPROG function is based on M.J.D. Powell’s implementation of the Goldfarb and Idnani dual quadratic programming (QP) algorithm for convex QP problems subject to general linear equality/inequality constraints (Goldfarb and Idnani 1983). That is, problems of the form:

subject to:

given the vectors b0, b1, and g, and the matrices H, A0, and A1. Matrix H is required to be positive definite. In this case, a unique x solves the problem, or the constraints are inconsistent. If H is not positive definite, a positive definite perturbation of H is used in place of H. For more details, see Powell (1983, 1985).

If a perturbation of H, H + αI, is used in the QP problem, H + αI also should be used in the definition of the Lagrange multipliers.


In this example, the QP problem:

min f(x) = –x20 + x21 + x22 + x23 + x24 – 2x1x2– 2x3x4 –2x0

subject to:

x0 + x1 + x2 + x3 + x4 = 5

x2 – 2x3 – 2x4 = –3

is solved.

RM, a, 2, 5
; Define the coefficient matrix A.
row 0: 1 1 1	1	1
row 1: 0 0 1 -2 -2
h = [[2,	0,	0,	0,	0], [0,	2, -2,	0,	0], $
  [0, -2,	2,	0,	0], [0,	0,	0,	2, -2], $
  [0,	0,	0, -2,	2]]
; Define the Hessian matrix of the objective function. Notice
; that since h is symmetric, the array concatenation operators
; “[ ]” are used to define it. b = [5, -3]
; Define b.
g = [ -2, 0, 0, 0, 0]
; Define g.
x = IMSL_QUADPROG(a, b, g, h)
PM, x




Warning Errors

MATH_NO_MORE_PROGRESS: Due to the effect of computer rounding error, a change in the variables fails to improve the objective function value. Usually, the solution is close to optimum.

Fatal Errors

MATH_SYSTEM_INCONSISTENT: System of equations is inconsistent. There is no solution.


Result = IMSL_QUADPROG(A, B, G, H [, DIAG=variable] [, /DOUBLE] [, DUAL=variable] [, MEQ=value] [, OBJ=variable])

Return Value

The solution x of the QP problem.



Two-dimensional matrix containing the linear constraints.


One-dimensional matrix of the right-hand sides of the linear constraints.


One-dimensional array of the coefficients of the linear term of the objective function.


Two-dimensional array of size N_ELEMENTS(g) x N_ELEMENTS(g) containing the Hessian matrix of the objective function. It must be symmetric positive definite. If h is not positive definite, the algorithm attempts to solve the QP problem with h replaced by h + Diag*I, such that h + Diag*I is positive definite.


DIAG (optional)

Name of the variable into which the scalar, equal to the multiple of the identity matrix added to h to give a positive definite matrix, is stored.

DOUBLE (optional)

If present and nonzero, then double precision is used.

DUAL (optional)

Name of the variable into which an array with N_ELEMENTS(g) elements, containing the Lagrange multiplier estimates, is stored.

MEQ (optional)

Number of linear equality constraints. If MEQ is used, then the equality constraints are located at a(i, *) for i = 0, ..., Meq – 1. Default: MEQ = N_ELEMENTS(a(*, 0)) n; i.e., all constraints are equality constraints.

OBJ (optional)

Name of variable into which optimal object function found is stored.

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